In recent work, Fomin et al. [1]} showed that \(c^k N^{O(1)}\)  time algorithms (\(c\in O(1)\) ) for \(\Phi \) -Extension lead to competitive exponential-time algorithms for many \(\Phi \) -Subset problems. The main tool was a simple randomized algorithm which solves \(\Phi \) -Subset in time \((2-\frac{1}{c})^n N^{O(1)}\)  if there is an algorithm that solves \(\Phi \) -Extension in time \(c^k N^{O(1)}\) . A derandomization was also given, turning the randomized algorithm into a deterministic one at the cost of a \(2^{o(n)}\)  factor in the running time. The method was also adapted to enumeration algorithms and combinatorial upper bounds.
This framework, together with a large body of work in parameterized algorithmics [2]}, where \(c^k N^{O(1)}\)  time algorithms are readily available for many subset problems, led to faster algorithms for around 30 decision and enumeration problems.
